Bayes' theorem
/ BAYZ /
Bayes' theorem is the rule for flipping a conditional probability around. You often know the chance of a symptom given a disease, but what you really want is the chance of the disease given the symptom. Bayes' theorem is the mathematical machine that turns the first into the second — it is the formal recipe for updating a belief when new evidence arrives.
In its simplest form, P(A | B) = P(B | A) × P(A) / P(B). You start with a prior belief P(A) — the chance of A before the evidence. The evidence B arrives, and you multiply by how likely that evidence is if A were true, then divide by how likely the evidence is overall, to get the posterior P(A | B). The famous, counter-intuitive lesson: a test that is 99% accurate for a disease that only 1 in 1000 people have will still produce mostly false alarms, because the rare disease is swamped by the many healthy people who occasionally test positive. The base rate matters enormously.
Bayes' theorem is the philosophical backbone of much actuarial reasoning. Bayesian credibility — blending a policyholder's own claims experience with the experience of the wider group — is Bayes' theorem in action, with the group rate as the prior and the individual's record as the evidence. It underlies disease-test interpretation in health actuarial work, fraud scoring, and any setting where you must rationally revise a probability as data accumulates.
Disease in 1/1000; test 99% accurate. Out of 1000 people, ~1 true case tests positive and ~10 healthy people also test positive. So a positive result means only about 1-in-11 chance of really being ill — the rare base rate dominates.
A very accurate test for a very rare condition still yields mostly false positives.
Ignoring the prior (base rate) is the classic 'base-rate fallacy'; Bayes' theorem only gives sensible answers when the prior is honestly included.